The natural carved turn radius of a ski is determined by the relation between its length and side cut, and the angle theta formed between the ski and the slope during turns. The turn radius is not constant for a given ski, but decreases with increasing theta angle. Mathematically, the nominal turning radius of a ski is defined by the following equation (from "Skiing Mechanics" by John Howe, 1983, p. 102): r=(L.sup.z Cos theta)/8 sc, where r is the natural carved turn radius of the ski, L is the length of the ski, sc is the side cut of the ski, and theta is as defined above. The natural turning radius of skis is ordinarily 150 to over 200 feet. A much shorter turning radius is desirable, particularly on steep slopes, because this is the dominant factor in control of speed and balance. The long turning radius of conventional skis leads to unacceptably high speeds during turns, even on moderate slopes. Only expert skiers on especially designed slopes can make true carved turns; most skiers maintain only imperfect control as a result of these long turning radii. In addition to the danger to themselves and others resulting from this instability, they must side-slip the back of their skis to shorten the turn and thereby reduce speed to maintain control. Side-slipping results in effective loss of a significant portion of the ski edges with resulting loss of supporting "platform". Side-slipping requires unweighting of the skis, a maneuver many skiers never adequately learn and which in any case requires much energy. On steeper slopes, this side-slipping results in the formation of moguls.
Another advantage of a short turning radius is that lower speeds are possible without dropping below the speed necessary to overcome critical angle effects encountered at the beginning of turns. The critical angle is defined as that angle of traverse below which the skier is unstable when gravitational forces in the plane of the slope equal or exceed the centrifugal force generated by a turn. Since centrifugal force varies directly with the square of the velocity and inversely with the radius of the turn, skis with shorter turning radii generate much greater centrifugal force at a given velocity. For example, skis with a natural turning radius of 45 feet require a velocity of only 15 MPH to maintain stability on a slope of 20 degrees at the beginning of a turn, as compared to 30 MPH on skis with a turning radius of 190 feet (these figures are derived from equation 10-2, page 121 of the reference cited above).
Snowboards, being shorter than skis, and wider (which allows for side cuts on the order of 0.8 inch or more), have natural carved turn radii on the order of 30 to 40 feet. Conventional skis, by contrast, are constrained by their current design to side cuts on the order of only 0.3 inch.